Finite Group Databasebeta

← #150002

Subgroups — #150002

Order 512.

311 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29145C211yesmaximal, characteristic
2#11529C211yesmaximal, characteristic
3#11475C211yesmaximal, characteristic
4#7367C214yesmaximal, complemented
5#2862C411yescharacteristic
6C4×#127C2211yescharacteristic
7#2023—21no—
8#2023—21no—
9#1961C412yes—
10#1836C2212yes—
11#1804—21no—
12#1804—21no—
13#1767C411yescharacteristic
14#1465—21no—
15#1459—21no—
16#1325C2212yes—
17#1325C2212yes—
18#1318—21no—
19#1318—21no—
20C2×C4⋊Q8—21no—
21C2×#127C2×C411yescharacteristic
22C2×C4×D4—41no—
23C2×C4×D4D411yescharacteristic
24C2×C4×D4D411yescharacteristic
25C2×C4⋊C8—21no—
26C2×#102—22no—
27C2×#102—22no—
28#407—41no—
29C2×C22⋊C8—41no—
30C2×#97C2×C412yes—
31#391—41no—
32#391—41no—
33#390—41no—
34#390D411yescharacteristic
35#390D411yescharacteristic
36#384C2311yescharacteristic
37#382—21no—
38#382C2×C412yes—
39C43C2×C411yescharacteristic
40#340—22no—
41#340—22no—
42#336—21no—
43#324—22no—
44#324—22no—
45#323—41no—
46#323—41no—
47C22×Q8—41no—
48C22×D4—41no—
49C22×D4C22⋊C411yescharacteristic
50C22×D4C22⋊C411yescharacteristic
51C23×C4—21no—
52C23×C4—21no—
53C2×M16—81no—
54C22×C8—22no—
55C22×C8—81no—
56C4⋊Q8—82no—
57#127—22no—
58#127—22no—
59C4×D4—41no—
60C4×D4—41no—
61C4×D4—41no—
62C4×D4—41no—
63C4×D4—84no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4C2×D411yescharacteristic
68C2×C4⋊C4C2×D411yescharacteristic
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—41no—
72C2×C22⋊C4—41no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42C22⋊C411yescharacteristic
76C2×C42C22⋊C411yescharacteristic
77C2×C42C22×C411yescharacteristic
78C4⋊C8—41no—
79C4⋊C8—41no—
80#102—82no—
81#102—82no—
82C22⋊C8—42no—
83C22⋊C8—42no—
84C22⋊C8—81no—
85C22⋊C8—81no—
86#97—24no—
87#95—42no—
88#95—41no—
89#95—41no—
90C24—21no—
91C24—21no—
92C2×Q8—82no—
93C2×Q8—82no—
94C2×Q8—82no—
95C2×D4—21no—
96C2×D4—21no—
97C2×D4—21no—
98C2×D4—41no—
99C2×D4—41no—
100C2×D4—41no—
101C2×D4—41no—
102C2×D4—44no—
103C2×D4—82no—
104C2×D4—82no—
105C2×D4#10112yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—42no—
115C22×C4—82no—
116C22×C4—82no—
117C22×C4#9911yescharacteristic
118C22×C4#9911yescharacteristic
119C22×C4C2×C22⋊C411yescharacteristic
120M16—162no—
121C2×C8—42no—
122C2×C8—42no—
123C2×C8—42no—
124C2×C8—82no—
125C2×C8—82no—
126C2×C8—162no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C4⋊C4—82no—
132C4⋊C4—82no—
133C22⋊C4—81no—
134C22⋊C4—81no—
135C22⋊C4—82no—
136C22⋊C4—82no—
137C42—21no—
138C42—21no—
139C42—21no—
140C42—41no—
141C42—42no—
142C42—81no—
143C42#13112yes—
144C23—22no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—81no—
151C23—81no—
152C23—81no—
153C23—81no—
154C23C2×#9911yescharacteristic
155Q8—164no—
156D4—42no—
157D4—42no—
158D4—81no—
159D4—81no—
160D4—88no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—161no—
182C2×C4—161no—
183C2×C4#35411yescharacteristic
184C2×C4#35411yescharacteristic
185C2×C4#40811yescharacteristic
186C2×C4C2×#10111yescharacteristic
187C8—82no—
188C8—162no—
189C8—162no—
190C22—21no—
191C22—21no—
192C22—42no—
193C22—82no—
194C22—82no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22—81no—
199C22—81no—
200C22—161no—
201C22—161no—
202C22#147211yescharacteristic
203C22C2×#35411yescharacteristic
204C22#213411yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—42no—
210C4—42no—
211C4—41no—
212C4—81no—
213C4—81no—
214C4—162no—
215C2—21no—
216C2—21no—
217C2—82no—
218C2—161no—
219C2#1156911yescharacteristic
220C2#1359511yescharacteristic
221C2#1359811yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7367C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#291452
#115292
#114752
#73672